The present invention relates to a new and distinct variety of flat peach tree, Prunus persica, which has been given the variety denomination ‘ASFPBF0798’. This new tree produces fruit with a long shelf life without alteration both on the tree after growth completion and after harvesting, very good eating quality, semi-clingstone white flesh fruit for fresh market in August in the Pyrénées-Orientales departement, France. Contrast is made to ‘Nectarmagie’ (U.S. Plant Pat. No. 17,579) white nectarine tree, ‘ASFPBF0492’ (U.S. Plant Pat. No. 21,143) white flat peach tree, ‘Maillarflat’ (Sweetcap) (non-patented) white flat peach tree, standard varieties, for reliable description. ‘ASFPBF0798’ is a promising candidate for commercial success in that it has very attractive fruits with very long shelf life without alteration before after harvesting.